1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integration by Parts - Choice of variables

  1. Feb 19, 2010 #1
    1. The problem statement, all variables and given/known data

    I'm getting different results when choosing my u & dv for Integration by Parts on the following integral:

    [tex]\int 2x^3 e^x^2 dx [/tex]
    (Note, the exponent on 'e' is x^2)

    This yields the correct solution:
    u = [tex]x^2[/tex]
    dv = [tex]2x e^x^2 dx [/tex]

    du = [tex]2xdx[/tex]
    v = [tex]e^x^2[/tex]

    However, I have tried using this instead (*)

    u = [tex]2x^3[/tex]
    dv = [tex]e^x^2 dx [/tex]

    du = [tex]6x^2 dx[/tex]
    v = [tex](e^x^2) / 2x[/tex]

    and this is yielding the incorrect solution (see 3.)

    2. Relevant equations
    Integration by Parts:
    [tex]\int udv = uv - \int vdu [/tex]

    3. The attempt at a solution

    The correct solution turns out to be
    [tex] x^2 e^x^2 - e^x^2 + C[/tex]

    When I use my other choice of variables (*), I get (using IBP)
    [tex] \int 2x^3 e^x^2 dx = 2x^3 e^x^2 / 2x - \int e^x^2 / 2x * 6x^2 dx [/tex]
    which leads to:
    [tex]x^2 e^x^2 - 3/2 e^x^2 + C [/tex]

    which is different from the other choice of variables.

    I've looked over both choices of variables, and I don't know why the second choice (*) comes up with a different solution.

    Thanks for the help!
  2. jcsd
  3. Feb 19, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Your error is that the integral of [tex]e^{x^2}[/tex] isn't [tex]e^{x^2}/(2x)[/tex]. (I'm assuming you meant to have parentheses on the bottom, but perhaps not. Regardless, the answer is incorrect either way.)
  4. Feb 19, 2010 #3
    Ah! Silly me, I've been staring at it and completely overlooked that. Thanks for your quick reply, I'll avoid such carelessness in the future :)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Integration Parts Choice Date
Integration by parts/substitution Nov 2, 2017
Solving an Integral Sep 23, 2017
Integration by parts problem Jul 18, 2017
Integration by Parts Twice Feb 21, 2017